3.3.86 \(\int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) [286]

3.3.86.1 Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac {64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}} \]

output

2/13*I*a*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^(1/2)+256/3003*I*a^4*sec(d*x+c) 
^7/d/(a+I*a*tan(d*x+c))^(7/2)+64/429*I*a^3*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c 
))^(5/2)+24/143*I*a^2*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^(3/2)
 

3.3.86.2 Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {2 \sec ^6(c+d x) (390 \cos (c+d x)+445 \cos (3 (c+d x))+7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))) (i \cos (4 (c+d x))+\sin (4 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{3003 d} \]

input

Integrate[Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]],x]
 

output

(2*Sec[c + d*x]^6*(390*Cos[c + d*x] + 445*Cos[3*(c + d*x)] + (7*I)*(26*Sin 
[c + d*x] + 59*Sin[3*(c + d*x)]))*(I*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])* 
Sqrt[a + I*a*Tan[c + d*x]])/(3003*d)
 

3.3.86.3 Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3975, 3042, 3975, 3042, 3975, 3042, 3974}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (c+d x)^7 \sqrt {a+i a \tan (c+d x)}dx\)

\(\Big \downarrow \) 3975

\(\displaystyle \frac {12}{13} a \int \frac {\sec ^7(c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \int \frac {\sec (c+d x)^7}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3975

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \int \frac {\sec ^7(c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \int \frac {\sec (c+d x)^7}{(i \tan (c+d x) a+a)^{3/2}}dx+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3975

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\sec ^7(c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\sec (c+d x)^7}{(i \tan (c+d x) a+a)^{5/2}}dx+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3974

\(\displaystyle \frac {12}{13} a \left (\frac {8}{11} a \left (\frac {8 i a^2 \sec ^7(c+d x)}{63 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\)

input

Int[Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]],x]
 

output

(((2*I)/13)*a*Sec[c + d*x]^7)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (12*a*((((2 
*I)/11)*a*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (8*a*((((8*I) 
/63)*a^2*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((2*I)/9)*a*S 
ec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(5/2))))/11))/13
 

3.3.86.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3974

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[2*b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^ 
(n - 1)/(f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] 
&& EqQ[Simplify[m/2 + n - 1], 0]
 

rule 3975

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n 
 - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1))   Int[(d*Se 
c[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, 
 m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m/2 + n - 1], 0] &&  !Inte 
gerQ[n]
 

3.3.86.4 Maple [A] (verified)

Time = 8.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {2 \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1024 i \cos \left (d x +c \right )+1024 \sin \left (d x +c \right )-128 i \sec \left (d x +c \right )+384 \sec \left (d x +c \right ) \tan \left (d x +c \right )-40 i \left (\sec ^{3}\left (d x +c \right )\right )+280 \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )-21 i \left (\sec ^{5}\left (d x +c \right )\right )+231 \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )\right )}{3003 d}\)	\(116\)

input

int(sec(d*x+c)^7*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

output

2/3003/d*(a*(1+I*tan(d*x+c)))^(1/2)*(1024*I*cos(d*x+c)+1024*sin(d*x+c)-128 
*I*sec(d*x+c)+384*sec(d*x+c)*tan(d*x+c)-40*I*sec(d*x+c)^3+280*tan(d*x+c)*s 
ec(d*x+c)^3-21*I*sec(d*x+c)^5+231*tan(d*x+c)*sec(d*x+c)^5)
 

3.3.86.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {128 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-429 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 286 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 104 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i\right )}}{3003 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

input

integrate(sec(d*x+c)^7*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
 

output

-128/3003*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-429*I*e^(6*I*d*x + 6 
*I*c) - 286*I*e^(4*I*d*x + 4*I*c) - 104*I*e^(2*I*d*x + 2*I*c) - 16*I)/(d*e 
^(12*I*d*x + 12*I*c) + 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c 
) + 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I*c) + 6*d*e^(2*I*d*x + 
 2*I*c) + d)
 

3.3.86.6 Sympy [F]

\[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \sec ^{7}{\left (c + d x \right )}\, dx \]

input

integrate(sec(d*x+c)**7*(a+I*a*tan(d*x+c))**(1/2),x)
 

output

Integral(sqrt(I*a*(tan(c + d*x) - I))*sec(c + d*x)**7, x)
 

3.3.86.7 Maxima [F(-1)]

Timed out. \[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \]

input

integrate(sec(d*x+c)^7*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
 

output

Timed out
 

3.3.86.8 Giac [F]

\[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{7} \,d x } \]

input

integrate(sec(d*x+c)^7*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
 

output

integrate(sqrt(I*a*tan(d*x + c) + a)*sec(d*x + c)^7, x)
 

3.3.86.9 Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.97 \[ \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{7\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,384{}\mathrm {i}}{11\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{13\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \]

input

int((a + a*tan(c + d*x)*1i)^(1/2)/cos(c + d*x)^7,x)
 

output

(exp(- c*1i - d*x*1i)*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + 
 d*x*2i) + 1))^(1/2)*128i)/(7*d*(exp(c*2i + d*x*2i) + 1)^3) - (exp(- c*1i 
- d*x*1i)*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1 
))^(1/2)*128i)/(3*d*(exp(c*2i + d*x*2i) + 1)^4) + (exp(- c*1i - d*x*1i)*(a 
 - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*384 
i)/(11*d*(exp(c*2i + d*x*2i) + 1)^5) - (exp(- c*1i - d*x*1i)*(a - (a*(exp( 
c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(13*d*(e 
xp(c*2i + d*x*2i) + 1)^6)